Unit lattices of $D_4$-quartic number fields with signature $(2,1)$

TL;DR

This work analyzes the unit-lattice shapes of $D_4$-quartic fields with signature $(2,1)$ by embedding unit groups via the logarithmic map into the plane moduli space ${\\mathcal S}_2$. It shows that every unit-shape point $p_L$ lies on the boundary of the standard fundamental domain and is necessarily transcendental, while also identifying precise algebraic limit points on the boundary (notably $i\\sqrt{3}$, $\\tfrac{1}{2}+\\tfrac{i\\sqrt{3}}{2}$, and $\\tfrac{1}{7}+\\tfrac{4i\\sqrt{3}}{7}$) and conjecturing that the full set of limit points equals the boundary. The paper combines regulator bounds (Silverman, strengthened by Akhtari–Vaaler) with explicit unit constructions (Stender, LPS, Nakamula) to obtain three convergent families of fields whose unit-shape points approach these limits, and proves that fixing a real quadratic subfield eliminates limit points in $\\Omega(D_4,2,1)$. Overall, the results illuminate the distribution and geometry of unit lattices in these number fields and propose a precise conjectural description of limit points along the boundary, with potential implications for inverse Galois questions and lattice-based cryptographic considerations.

Abstract

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$.

Figures (6)

• Figure 1: Each plotted point corresponds to the shape of the unit lattice of a $D_4$-quartic field with signature $(2,1)$. We list all such fields of absolute discriminant $\leq 10^6$. The points appear to be dense in the boundary of the fundamental domain of $\mathop{\mathrm{GL}}\nolimits_2(\mathbb{Z}) \backslash \mathop{\mathrm{\mathfrak{h}}}\nolimits$.
• Figure 2:
• Figure 3:
• Figure 4: The gray region is the standard fundamental domain ${\mathscr{F}}$ of $\mathop{\mathrm{GL}}\nolimits_2(\mathbb{Z}) \backslash \mathop{\mathrm{\mathfrak{h}}}\nolimits$ in the complex plane.
• Figure 5: Each point of ${\mathscr{F}}$ corresponds to an equivalence class of rank $2$ lattices up to homothety and reflection. The groups are the automorphism groups of the corresponding lattices; for example, the cuspidal point $i$ corresponds to a lattice with automorphism group $D_4$. A lattice on the interior of ${\mathscr{F}}$ has only one nontrivial automorphism, given by multiplication by $-1$.